A c0-saturated banach space with no long unconditional basic sequences

Abstract

We present a Banach space X with a Schauder basis of length ω1 which is saturatedby copies of c0 and such that for every closed decomposition of a closedsubspace X = X0⊕ X1, either X0 or X1 has to be separable. This can be considered as the non-separable counterpart of the notion of hereditarily indecomposable space. Indeed, the subspaces of X have "few operators" in the sense that every bounded operator T: R → X from a subspace X of X into X is the sum of a multiple of the inclusion and a ω1- singular operator, i.e., an operator S which is not an isomorphism on any non-separable subspace of X. We also show that while X is not distortable (being c0-saturated), it is arbitrarily ω1-distortable in the sense that for every λ > 1 there is an equivalent norm |||. ||| on X such that for every non-separable subspace X of X there exist x, y ∈ SX suchthat |||x|||/|||y||| ≥ λ.